Linear Flavor-Wave Theory for Fully Antisymmetric $\mathrm{SU}(N)$ Irreducible Representations

TL;DR

This paper extends linear flavor-wave theory to fully antisymmetric $ ext{SU}(N)$ irreps to analyze magnetic order in various 2D lattices, revealing how quantum fluctuations influence the stability of color order.

Contribution

It introduces two methods for $ ext{SU}(N)$ flavor-wave analysis and compares their effectiveness in predicting magnetic order in different lattice geometries.

Findings

Long-range Néel order likely for $ ext{SU}(4)$ on square lattice with two particles per site.

Quantum fluctuations tend to destroy order for larger $N$ and more particles per site.

Color order is unlikely on triangular and honeycomb lattices for $m>1$, except marginally on honeycomb with $m=2$.

Abstract

The extension of the linear flavor-wave theory (LFWT) to fully antisymmetric irreducible representations (irreps) of $\mathrm{SU}(N)$ is presented in order to investigate the color order of $\mathrm{SU}(N)$ antiferromagnetic Heisenberg models in several two-dimensional geometries. The square, triangular and honeycomb lattices are considered with $m$ fermionic particles per site. We present two different methods: the first method is the generalization of the multiboson spin-wave approach to $\mathrm{SU}(N)$ which consists of associating a Schwinger boson to each state on a site. The second method adopts the Read and Sachdev bosons which are an extension of the Schwinger bosons that introduces one boson for each color and each line of the Young tableau. The two methods yield the same dispersing modes, a good indication that they properly capture the semi-classical fluctuations, but the first one leads to spurious flat modes of finite frequency not present in the second one. Both methods lead to the same physical conclusions otherwise: long-range N\'eel-type order is likely for the square lattice for $\mathrm{SU}(4)$ with two particles per site, but quantum fluctuations probably destroy order for more than two particles per site, with $N=2m$. By contrast, quantum fluctuations always lead to corrections larger than the classical order parameter for the tripartite triangular lattice (with $N=3m$) or the bipartite honeycomb lattice (with $N=2m$) for more than one particle per site, $m>1$, making the presence of color very unlikely except maybe for $m=2$ on the honeycomb lattice, for which the correction is only marginally larger than the classical order parameter.